CHAPTER 1

Introduction

Let K be an obstacle in Rn (n ≥ 3, n odd), i.e. a compact subset of Rn with

C∞ boundary ∂K such that ΩK = Rn \ K is connected. One of the main objects

of study in the classical scattering theory (by an obstacle) is the so called scattering

matrix S(z) related to the wave equation in R×Ω with Dirichlet boundary condition

on R × Ω. This is a meromorphic operator-valued function

S(z) :

L2(Sn−1)

−→

L2(Sn−1)

with poles (resonances) {λj}j=1

∞

in the half-plane Im(z) 0 (see [LP1], [M2]

or [Z1]). The resonances can also be defined as the poles of the meromorphic

continuation of the cut-off resolvent of the self-adjoint realization in L2(Rn \ K) of

the Laplacian −∆ with Dirichlet boundary conditions.

A variety of problems in scattering theory deal with extracting geometric in-

formation about K from the distribution of the poles {λj}. In what follows we

describe one particular problem of this kind.

The obstacle K is called trapping if there exists an infinitely long bounded

broken geodesic (in the sense of Melrose and Sj¨ostrand [MS]) in the exterior domain

Ω. It follows from results of Lax-Phillips [LP2] (see also Vainberg [Va] and Melrose-

Sj¨ ostrand [MS]) that if K is non-trapping, then {z ∈: 0 Im(z) α} contains

finitely many poles λj for any α 0 (cf. the Epilogue in [LP1] for more precise

information). In the first edition of their monograph Scattering Theory published

in 1967, Lax and Phillips conjectured that for trapping obstacles there should exist

a sequence {λj} of scattering poles such that Imλj → 0 as j → ∞. However

M. Ikawa [I1] showed that this is not the case when K is a disjoint union of two

strictly convex compact domains with smooth boundaries. It turns out that in this

particular case the scattering matrix has poles approximately at the points

kπ

d

+iδ,

k = 0, ±1,±2, . . ., where d is the distance between the two connected components

K1 and K2 of K and δ 0 is a constant depending only on the curvatures of

∂K at the ends of the shortest segment connecting K1 and K2. Substantial new

information concerning the distribution of poles in this case was later given by C.

Gerard [G].

Ikawa modified the initial conjecture of Lax and Phillips in the following way.

Modified Lax-Phillips Conjecture (MLPC): If K is trapping, then there

exists α 0 such that the strip {z : 0 Im(z) α} contains infinitely many

scattering resonances λj .

By now a lot of results have been obtained on distribution of resonances in

various aspects of scattering theory. We refer the reader to the monograph [M2] of

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